{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "read \"contiguous2f1 .mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnThis~is~a~Maple~package~ for~computing~contiguous~relations~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%WGauss~hypergeometric~2F1~series,~written~by~R.~VidunasG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%>~Version~3.20,~April~29,~2002G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 50 "Th e current reference functions are the following:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*HypergeomG6%7$%\"aG%\"bG7#%\"cG%\"zG-F$6%7$,&F'\"\" \"F0F0F(F)F+" }}{PARA 6 "" 1 "" {TEXT -1 58 " You can change them with commando 'reference2f1(a,b,c,z)'" }}{PARA 6 "" 1 "" {TEXT -1 50 " The help function is envoked by 'contig2f1help()'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "contig2f1he lp( );" }}{PARA 6 "" 1 "" {TEXT -1 54 " This is a Maple 7.0 package f or computing contiguous" }}{PARA 6 "" 1 "" {TEXT -1 50 " relation s of Gauss hypergeometric 2F1 series" }}{PARA 6 "" 1 "" {TEXT -1 42 " \+ Version 3.20, April 29, 2002" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 57 "(c) Raimundas Vidunas, University \+ of Amsterdam, 2000-2001" }}{PARA 6 "" 1 "" {TEXT -1 47 " \+ Antwerp University, 2002" }}{PARA 6 "" 1 "" {TEXT -1 53 "Supp orted by the Dutch NWO, project number 613-06-565" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "By definition, two 2F1 hy pergeometric series are contiguous when" }}{PARA 6 "" 1 "" {TEXT -1 56 "their corresponding upper and lower parameters differ by" }}{PARA 6 "" 1 "" {TEXT -1 63 "integers, see G.E.Andrews, R.Askey, R.Roy, \"Sp ecial Functions\"," }}{PARA 6 "" 1 "" {TEXT -1 52 "Cambridge Univ. Pre ss, Cambridge, 1999; chapter 2.5." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 53 "For any three contiguous 2F1 hypergeometr ic functions" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%-%*HypergeomG6%7$,&%\" aG\"\"\"&%\"kG6#F)F),&%\"bGF)&%\"lGF,F)7#,&%\"cGF)&%\"mGF,F)%\"zG-F$6% 7$,&F(F)&F+6#\"\"#F),&F.F)&F0F " 0 "" {MPLTEXT 1 0 31 "contig2f1help( contiguous2f1 );" }}{PARA 6 "" 1 "" {TEXT -1 59 " Given three triples of integers k[i], l[i], m[i] (i=1,2,3)," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%.Contiguous2f1G6%7%&%\"kG6#\"\"\"&%\"lGF)& %\"mGF)7%&F(6#\"\"#&F,F1&F.F17%&F(6#\"\"$&F,F7&F.F7" }}{PARA 6 "" 1 " " {TEXT -1 29 "returns a contiguous relation" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/,(*&%\"AG\"\"\"-%*HypergeomG6%7$,&%\"aGF'&%\"kG6#F'F', &%\"bGF'&%\"lGF0F'7#,&%\"cGF'&%\"mGF0F'%\"zGF'F'*&%\"BGF'-F)6%7$,&F-F' &F/6#\"\"#F',&F2F'&F4FBF'7#,&F7F'&F9FBF'F:F'F'*&%\"CGF'-F)6%7$,&F-F'&F /6#\"\"$F',&F2F'&F4FPF'7#,&F7F'&F9FPF'F:F'F'\"\"!%!G" }}{PARA 6 "" 1 " " {TEXT -1 65 "whereas function 'contiguous2f1' with the same argument s computes" }}{PARA 6 "" 1 "" {TEXT -1 57 "the coefficient list [A,B,C ] of this contiguous relation." }}{PARA 6 "" 1 "" {TEXT -1 55 "The ref erence function Hypergeom([a,b],[c],z) is set by" }}{PARA 6 "" 1 "" {TEXT -1 41 "the commando 'reference2f1([a,b],[c],z)'." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 55 "Both functions can b e called with one or two arguments." }}{PARA 6 "" 1 "" {TEXT -1 30 "Th e default last arguments are" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#3/7%&% \"kG6#\"\"#&%\"lGF(&%\"mGF(7%\"\"\"\"\"!F0/7%&F'6#\"\"$&F+F4&F-F47%F0F 0F0" }}{PARA 6 "" 1 "" {TEXT -1 63 "The arguments of function 'Contigu ous2f1' can be hypergeometric" }}{PARA 6 "" 1 "" {TEXT -1 51 "function s contiguous to the reference function, say" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#5-%*HypergeomG6%7$,&%\"aG\"\"\"\"\"#F*%\"bG7#,&%\"cGF*F *!\"\"%\"zG-%*hypergeomG6%7$F,F(F-F1" }}{PARA 6 "" 1 "" {TEXT -1 20 "i nstead of [2,0,-1]." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 56 "In general, the resulting contiguous relation is unique ." }}{PARA 6 "" 1 "" {TEXT -1 61 "However, in some special cases (when z=0 or z=1, and/or a, b," }}{PARA 6 "" 1 "" {TEXT -1 64 "c-a or c-b a re integers), the contiguous relation is not unique." }}{PARA 6 "" 1 " " {TEXT -1 60 "The returned result may not be the most wanted, for exa mple," }}{PARA 6 "" 1 "" {TEXT -1 62 "a zero coefficient may appear in an undesired place. Note also" }}{PARA 6 "" 1 "" {TEXT -1 58 "that if c is an integer, some hypergeometric series may be" }}{PARA 6 "" 1 " " {TEXT -1 61 "undefined, and contiguous relations would not make senc e. See" }}{PARA 6 "" 1 "" {TEXT -1 63 "contig2f1help(reference2f1) for more info on exceptional cases." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "contig2f1help( reference2f1 \+ );" }}{PARA 6 "" 1 "" {TEXT -1 47 "The commando 'reference2f1([a,b], [ c], z)' sets" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#3-%*HypergeomG6%7$%\"a G%\"bG7#%\"cG%\"zG-F%6$7$,&F(\"\"\"F1F1F)F," }}{PARA 6 "" 1 "" {TEXT -1 65 "as reference functions for commands Contiguous2f1, contiguous2f 1," }}{PARA 6 "" 1 "" {TEXT -1 62 "Contig2f1express, contig2f1express, contig2f1P and contig2f1Q." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 54 "Some reference functions may not express all fu nctions" }}{PARA 6 "" 1 "" {TEXT -1 61 "contiguous to them, contiguous relations with them may not be" }}{PARA 6 "" 1 "" {TEXT -1 64 "unique , and/or some contigous functions may be undefined. In the" }}{PARA 6 "" 1 "" {TEXT -1 61 "following three situations 'reference2f1' produce s a warning." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 55 "If z=0 or z=1, the contiguous relations are not unique." }} {PARA 6 "" 1 "" {TEXT -1 64 "'contiguous2f1' finds just one contiguous relation, perhaps with" }}{PARA 6 "" 1 "" {TEXT -1 60 "a zero coeffic ient. The commando 'contig2f1express' may fail" }}{PARA 6 "" 1 "" {TEXT -1 61 "to express a given contiguous 2F1 in the reference functi ons." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "I f a, b, c-a or c-b are integer(s), the contiguous relations may" }} {PARA 6 "" 1 "" {TEXT -1 64 "not unique (for example, for terminating \+ series), and contiguous" }}{PARA 6 "" 1 "" {TEXT -1 61 "expression of \+ a contiguous 2F1 in the reference functions may" }}{PARA 6 "" 1 "" {TEXT -1 56 "not exist (for example, when the reference functions are " }}{PARA 6 "" 1 "" {TEXT -1 51 "terminating, and the third one does n ot terminate)." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 62 "When c is an integer, some contiguous hypergeometric func tions" }}{PARA 6 "" 1 "" {TEXT -1 57 "(with non-positive lower paramet er) are not well-defined." }}{PARA 6 "" 1 "" {TEXT -1 57 "Then the ret urned contiguous relation may be meaningless." }}{PARA 6 "" 1 "" {TEXT -1 62 "If for those functions an upper parameter is a non-positi ve as" }}{PARA 6 "" 1 "" {TEXT -1 64 "well, and can be interpreted as \+ terminating series, the returned" }}{PARA 6 "" 1 "" {TEXT -1 51 "conti guous relation relation is (accordingly) good." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 59 "Contiguous relations may \+ not be unique in other cases also," }}{PARA 6 "" 1 "" {TEXT -1 63 "for example, when z=-1 and c-a+b is an integer (recall Kummer's" }}{PARA 6 "" 1 "" {TEXT -1 64 "formula). However, most contiguous relations ar e unique in these" }}{PARA 6 "" 1 "" {TEXT -1 61 "other cases, includi ng the one implied by 'contig2f1express'." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "contig2f1help( Conti g2f1express );" }}{PARA 6 "" 1 "" {TEXT -1 64 "The commandos 'contig2f 1P(k,l,m)' and 'contig2f1Q(k,l,m)' return" }}{PARA 6 "" 1 "" {TEXT -1 58 "the rational coefficients P and Q in a contiguous relation" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*HypergeomG6%7$,&%\"aG\"\"\"%\"kGF* ,&%\"bGF*%\"lGF*7#,&%\"cGF*%\"mGF*%\"zG,&*&%\"PGF*-F%6%7$F)F-7#F1F3F*F **&%\"QGF*-F%6%7$,&F)F*F*F*F-F:F3F*F*" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 55 "The reference function Hypergeom([a,b ],[c],z) is set by" }}{PARA 6 "" 1 "" {TEXT -1 63 "the commando 'refer ence2f1([a,b],[c],z)'. The returned relation" }}{PARA 6 "" 1 "" {TEXT -1 62 "is the specialization of the corresponding contiguous relation " }}{PARA 6 "" 1 "" {TEXT -1 59 "general a, b, c, z. Check 'contig2f1h elp(reference2f1)' for" }}{PARA 6 "" 1 "" {TEXT -1 42 "the information on exceptional situations." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 61 "The commando 'Contig2f1express(k,l,m)' returns \+ the right-hand" }}{PARA 6 "" 1 "" {TEXT -1 62 "of the above expression , and 'contig2f1express(k,l,m)' returns" }}{PARA 6 "" 1 "" {TEXT -1 60 "the coefficient list [contig2f1P(k,l,m), contig2f1Q(k,l,m)]." }} {PARA 6 "" 1 "" {TEXT -1 60 "Function 'Contig2f1express' can be called with one argument," }}{PARA 6 "" 1 "" {TEXT -1 62 "which must be then a hypergeometric function contiguous to the" }}{PARA 6 "" 1 "" {TEXT -1 33 "reference functions; for example," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%1Contig2f1expressG6#-%*hypergeomG6%7$,&%\"aG\"\"\"%\"kGF,,&%\" bGF,%\"lGF,7#,&%\"cGF,%\"mGF,%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "contig2f1help( conti g2f1subs );" }}{PARA 6 "" 1 "" {TEXT -1 63 "If L is a list of three po lynomials, 'contig2f1subs(L)' returns" }}{PARA 6 "" 1 "" {TEXT -1 64 " the same list, but each polynomial is divided by the common GCD." }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 60 "If the co mmando 'contig2f1subs' has more arguments, the last" }}{PARA 6 "" 1 " " {TEXT -1 59 "argument is expected to be a list of three polynomials, and" }}{PARA 6 "" 1 "" {TEXT -1 64 "other arguments are interpreted a s substitutions. Simplification" }}{PARA 6 "" 1 "" {TEXT -1 59 "of GCD 's is performed after each (consequent) substitution." }}{PARA 6 "" 1 "" {TEXT -1 63 "The function is useful in manipulating output of conti guous2f1." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 15 "General a, b, c" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "c ontiguous2f1([1,0,0],[0,0,0],[-1,0,0]);\nContiguous2f1([1,0,0],[0,0,0] ,[-1,0,0]);\nContiguous2f1( Hypergeom([b,a-1],[c],z ) );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%,$*&,&%\"zG\"\"\"F(!\"\"F(%\"aGF(F),*F*!\"#%\" cGF(*&F'F(F*F(F(*&%\"bGF(F'F(F),&F-F)F*F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"zG\"\"\"F(!\"\"F(%\"aGF(-%*HypergeomG6%7$,&F* F(F(F(%\"bG7#%\"cGF'F(F)*&,*F*!\"#F2F(*&F'F(F*F(F(*&F0F(F'F(F)F(-F,6%7 $F*F0F1F'F(F(*&,&F2F)F*F(F(-F,6%7$,&F*F(F(F)F0F1F'F(F(\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"cG!\"\"%\"aG\"\"\"F*-%*HypergeomG6 %7$%\"bG,&F)F*F*F(7#F'%\"zGF*F**(,&F2F*F*F(F*F)F*-F,6%7$,&F)F*F*F*F/F1 F2F*F(*&,*F)!\"#F'F**&F2F*F)F*F**&F/F*F2F*F(F*-F,6%7$F)F/F1F2F*F*\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "contiguous2f1([1,0,0],[ 0,0,-2]);\nContiguous2f1([0,0,0],[1,0,0],[0,0,-2]);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#7%*&,.%\"cG\"\"\"\"\"#!\"\"*(F(F'F&F'%\"zGF'F)*&\"\"$ F'F+F'F'*&F+F'%\"aGF'F'*&%\"bGF'F+F'F'F'F/F'*(,&F+F'F'F)F',&F&F'F(F)F' ,&F&F'F'F)F',$*&,,*&F&F'F+F'F)*&F(F'F+F'F'F&F'F(F)F.F'F',(F&F)F/F'F'F' F'F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(,,*&%\"cG\"\"\"%\"zGF)!\" \"*&\"\"#F)F*F)F)F(F)F-F+*&F*F)%\"aGF)F)F),(F(F+F/F)F)F)F)-%*Hypergeom G6%7$F/%\"bG7#F(F*F)F+*(,.F(F)F-F+*(F-F)F(F)F*F)F+*&\"\"$F)F*F)F)F.F)* &F5F)F*F)F)F)F/F)-F26%7$,&F/F)F)F)F5F6F*F)F)**,&F*F)F)F+F),&F(F)F-F+F) ,&F(F)F)F+F)-F26%F47#FCF*F)F)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Co ntiguous functions to Kummer's formula" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "reference2f1(a-1,b,a-b,-1);" }}{PARA 6 "" 1 "" {TEXT -1 50 "The current reference functions are the following:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*HypergeomG6%7$,&%\"aG\"\"\"F)!\"\"%\"bG7#,&F (F)F+F*F*-F$6%7$F(F+F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "contiguous2f1([2,0,0]);\ncontig2f1subs(b=a, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$%\"aG\"\"#,$F%!\"#%\"bG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#!\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Contig2f1express( 2,0,0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"bG\"\"\"%\"aG!\"\"-%*HypergeomG6%7$,&F'F&F&F(F%7 #,&F'F&F%F(F(F&#F(\"\"#-F*6%7$F'F%F.F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "contiguous2f1([0,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$%\"bG!\"\",&%\"aG\"\"\"F)F&,(F(F&F)F)F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "contig2f1express(0,0,1);\nContig2f1expres s(0,0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$**,&%\"aG\"\"\"%\"bG! \"\"F(,(F'\"\"#*&F,F(F)F(F*F(F*F(,&F'F(*&F,F(F)F(F*F*,&F)F(F(F*F*F*,$* *,&F'F(F(F*F(F&F(F0F*F.F*F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*,,&% \"aG\"\"\"%\"bG!\"\"F',(F&\"\"#*&F+F'F(F'F)F'F)F',&F&F'*&F+F'F(F'F)F), &F(F'F'F)F)-%*HypergeomG6%7$,&F&F'F'F)F(7#F%F)F'F)*.F+F'F4F'F%F'F/F)F- F)-F16%7$F&F(F5F)F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "co ntiguous2f1( [0,0,1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&,&%\" aG\"\"\"*&\"\"#F(%\"bGF(!\"\"F(,&F+F(F(F,F(F,,$*&,&F'F(F(F,F(,&F'F(F+F ,F(F*,$*&F1F(,(F'F**&F*F(F+F(F,F(F,F(F," }}}{PARA 0 "" 0 "" {TEXT -1 36 "Contiguous to a logarithmic function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "reference2f1( 1, 1, 3, z );" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, Terminating 2F1's do not express nonterminating series \n" }}{PARA 7 "" 1 "" {TEXT -1 74 "Warning, 2F1's with lower parameter a negative integer may not be defined\n" }}{PARA 6 "" 1 "" {TEXT -1 50 "The current reference functions are the following:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*HypergeomG6%7$\"\"\"F'7#\"\"$%\"zG-F$6%7$\"\"#F 'F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Contiguous2f1( [1, 2, 4] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"zG\"\"%\"\"\"-%* HypergeomG6%7$\"\"#\"\"$7#\"\"(F'F)!\"#*&,*\"$]\"F)*&\"$v\"F))F'F.F)F) *&\"$5$F)F'F)!\"\"*&\"#:F))F'F/F)F;F)-F+6%7$F.F)7#F/F'F)F)*&,(!$]\"F)* &\"$0\"F)F8F)F;*&\"$g#F)F'F)F)F)-F+6%7$F)F)FBF'F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Contiguous2f1( Hypergeom( [1,2], [4 ], z ) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*&%\"zG\"\"\"-%*Hyperg eomG6%7$F'\"\"#7#\"\"%F&F'!\"#*&,&F&\"\"$F2!\"\"F'-F)6%7$F,F'7#F2F&F'F '*&F2F'-F)6%7$F'F'F7F&F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "20 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }